Index: The Book of Statistical ProofsProbability Distributions ▷ Multivariate continuous distributions ▷ Normal-gamma distribution ▷ Covariance

Theorem: Let $x \in \mathbb{R}^n$ and $y > 0$ follow a normal-gamma distribution:

$\label{eq:ng} x,y \sim \mathrm{NG}(\mu, \Lambda, a, b) \; .$

Then,

1) the covariance of $x$, conditional on $y$ is

$\label{eq:ng-cov-cond} \mathrm{Cov}(x \vert y) = \frac{1}{y} \Lambda^{-1} \; ;$

2) the covariance of $x$, unconditional on $y$ is

$\label{eq:ng-cov-x} \mathrm{Cov}(x) = \frac{b}{a-1} \Lambda^{-1} \; ;$

3) the variance of $y$ is

$\label{eq:ng-var-y} \mathrm{Var}(y) = \frac{a}{b^2} \; .$

Proof:

1) According to the definition of the normal-gamma distribution, the distribution of $x$ given $y$ is a multivariate normal distribution:

$\label{eq:ng-mvn} x \vert y \sim \mathcal{N}(\mu, (y \Lambda)^{-1}) \; .$ $\label{eq:mvn-cov} x \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad \mathrm{Cov}(x) = \Sigma \; ,$

such that we have:

$\label{eq:ng-cov-cond-qed} \mathrm{Cov}(x \vert y) = (y \Lambda)^{-1} = \frac{1}{y} \Lambda^{-1} \; .$

2) The marginal distribution of the normal-gamma distribution with respect to $x$ is a multivariate t-distribution:

$\label{eq:ng-marg-x} x \sim t\left( \mu, \left(\frac{a}{b} \Lambda \right)^{-1}, 2a \right) \; .$ $\label{eq:mvt-cov} x \sim t(\mu, \Sigma, \nu) \quad \Rightarrow \quad \mathrm{Cov}(x) = \frac{\nu}{\nu-2} \Sigma \; ,$

such that we have:

$\label{eq:ng-cov-x-qed} \mathrm{Cov}(x) = \frac{2a}{2a-2} \left(\frac{a}{b} \Lambda \right)^{-1} = \frac{a}{a-1} \, \frac{b}{a} \, \Lambda^{-1} = \frac{b}{a-1} \Lambda^{-1} \; .$

3) The marginal distribution of the normal-gamma distribution with respect to $y$ is a univariate gamma distribution:

$\label{eq:ng-marg-y} y \sim \mathrm{Gam}(a, b) \; .$ $\label{eq:gam-var} x \sim \mathrm{Gam}(a, b) \quad \Rightarrow \quad \mathrm{Var}(x) = \frac{a}{b^2} \; ,$

such that we have:

$\label{eq:ng-var-y-qed} \mathrm{Var}(y) = \frac{a}{b^2} \; .$
Sources:

Metadata: ID: P345 | shortcut: ng-cov | author: JoramSoch | date: 2022-09-22, 09:17.